Let f( x ) be an indefinite integral of ^2 x . Consider the following statements : Statement 1 : The function f( x ) satisfies f( x + )=f( x ) for all real x . Statement 2: ^2(x+ )= ^2 x for all real x . Which one of the following is correct in respect of the above statements?

Step-by-step solution

aligned & f(x)= ^2 x d x= 1- 2 x 2 d x & = 1 2 x- 2 x 2 1 2 +c= 1 2 x- 1 4 2 x+c aligned (i) aligned & f( +x)= 1 2 ( +x)- 1 4 2( +x) & = 1 2 + 1 2 x- 1 2 (2 +2 x)+c & = x 2 - 1 2 2 x+c=f(x) aligned So, statement 1 is true. (ii) aligned & ^2( +x)= ^2 x & (- x)^2= ^2 x & ^2 x= ^2 x aligned So, statement 2 is true